Hyperbolic band topology with non-trivial second Chern numbers

Topological band theory establishes a standardized framework for classifying different types of topological matters. Recent investigations have shown that hyperbolic lattices in non-Euclidean space can also be characterized by hyperbolic Bloch theorem. This theory promotes the investigation of hyperbolic band topology, where hyperbolic topological band insulators protected by first Chern numbers have been proposed. Here, we report a new finding on the construction of hyperbolic topological band insulators with a vanished first Chern number but a non-trivial second Chern number. Our model possesses the non-abelian translational symmetry of {8,8} hyperbolic tiling. By engineering intercell couplings and onsite potentials of sublattices in each unit cell, the non-trivial bandgaps with quantized second Chern numbers can appear. In experiments, we fabricate two types of finite hyperbolic circuit networks with periodic boundary conditions and partially open boundary conditions to detect hyperbolic topological band insulators. Our work suggests a new way to engineer hyperbolic topological states with higher-order topological invariants.

) correspond to annihilation (creation) operators of four sublattices in the unit cell marked by (i, j, k, l), which quantify the implementation times of group operators 1 , 2 , 3 , 4 on the central unit. ) equal to m-a, m+a, -m+a, -m-a, respectively. The inter-cell coupling strengths along the hyperbolic translation direction marked by are described by ± (with j=1, 2, 3, 4), ± (with j=1, 3) and ± (with j= 2, 4), respectively. Following the hyperbolic band theory, the above lattice model in 2D hyperbolic space can be described in the momentum space. In particular, we can equip inter-cell couplings of the hyperbolic unit cell with twisted boundary conditions along four translation directions, where the phase factors (j=1, 2, 3, 4) along directions given by 1 , 2 , 3 and 4 are introduced. The phase factors related to their inverses are in the form of − (j=1, 2, 3, 4). In this case, four wave-vectors 1 , 2 , 3 and 4 can be regarded as Bloch vectors in 4D momentum space.

Supplementary Note 2. Periodic boundary conditions of the abelian and non-abelian clusters.
In this part, we illustrate the detailed boundary connection to construct the PBC for both abelian and non-abelian clusters. Before the discussion on finite hyperbolic clusters, we start to illustrate the general process to produce our designed hyperbolic lattice model. Based on the crystallography of {8,8} hyperbolic lattices, it has been pointed out that the {8, 8} hyperbolic lattice can be constructed by applying translational operations (generated by four generators of 1 , 2 , 3 and 4 ) to the unit cell. The representation of these generators in the Poincaré disk can be expressed as = (( − 1) ) 1 (−( − 1) ) with = 1,2,3,4. Here, R is the rotation matrix written by where ∪ denotes disjoint union, and the set of coset representatives is The cluster with N units (C) could be generate by the Bolza cell (D) with the group of T, that is representation of a group element of is expressed as: Based on the N×N matrix representation of the Supplementary equation (8) Based on the matrix representations, it is straightforwardly to know that the four matrix representations of abelian cluster (Supplementary equation (9)) satisfy the relationship of (γ ) (γ ) = (γ ) (γ ). While, as for that of non-abelian cluster, these four matrix representations (Supplementary equation (6)) obey (γ ) (γ ) ≠ (γ ) (γ ). These results clearly prove the abelian and non-abelian nature of our proposed hyperbolic clusters.
For a finite abelian PBC cluster, it has been demonstrated that the hyperbolic crystal momentum becomes discretized. The discretized k-vector in 4D BZ can be obtain by simultaneous diagonalization of the translation matrices of four group generators, as expressed in Supplementary equation (9). In this case, the N allowed k values correspond to N eigenvalues of these four simultaneously diagonalized matrices. Substituting the obtained k vectors into the Bloch Hamiltonian of the non-trivial hyperbolic model, we can calculate the eigen-spectra of the abelian cluster, as shown in Figures 2a and 2b of main text.

Supplementary Note 4. Numerical results of the eigen-spectra for non-abelian clusters with
PBCs. In this part, we numerically calculate the eigen-spectra of finite non-abelian cluster with PBCs by the direct diagonalization. The calculated eigen-spectra with different mass terms (m=0.7, a=0.2) and (m=0.7, a=3.2) are shown in Supplementary Figures 2a and 2b, respectively. Comparing to counterparts of the abelian cluster, we can see that the significant difference of energy spectra exists between abelian and non-abelian clusters with different mass terms. This indicates that the non-abelian cluster could not be described by the U(1) hyperbolic band theory of the topological Bolza cell.

Supplementary Figure 2. Numerical results of the eigen-spectra for non-abelian clusters with
PBCs. The calculated eigen-spectra of periodic non-abelian cluster with the mass term being (m=0.7, a=0.2) for (a) and (m=0.7, a=3.2) for (b).

Supplementary Note 5. Numerical results of the eigen-spectra for abelian clusters with fully
OBCs. In this part, we numerically calculate the eigen-spectra of finite abelian cluster with fully OBCs. The calculated eigen-spectra with different mass terms (m=0.7, a=0.2) and (m=0.7, a=3.2) are shown in Supplementary Figures 3a and 3b, respectively. The colormap corresponds to the quantity ( ), which quantifies the localization degree of each eigenmode on boundary sites. We can see that much of eigen-modes exhibit significant boundary localizations, making the bulk modes deviate from that of the periodic hyperbolic cluster. Hence, the nontrivial boundary states are hard to be resolved.

Supplementary Figure 3. Numerical results of the eigen-spectra for abelian clusters with fully
OBCs. The calculated eigen-spectra with the mass term being (m=0.7, a=0.2), for (a) and (m=0.7, a=3.2) for (b).